APPLIED MECHANICa 



[BEVIL OKARINO RACKS. 



cal surface*, of which is the apex for the one wheel, and 

 B the apex for the other ; and if from the centres C and 

 B the circlet E S and E T be described, they will repre- 

 sent the outline* of the developed surfaces of the cones 

 F C K. 1 > I 1 . K respectively, and become the pitch circles 

 on which the outlines of the teeth at E may be described. 

 we to cut these teeth in Pper, and then wrap 

 them round the cones FOE and D B E, we should have 

 them interlacing and gearing into each other at E. Pro- 

 ceeding in the same manner at the point M, by drawing 

 U V through M perpendicular to A E, and describing 

 from centres C and B the pitch circles W and X, with 

 radii respectively equal to V M and U M, we got the 

 development of the teeth at M, which are precisely simi- 

 lar to those at E, but on a smaller scale, their outlines 

 being denned by the radii converging from those in S and 

 T towards C and B, the centres of development. 



Bevil gearing applies when the motion ia to be cou- 

 Flf.lU. 



veyed at any rate of speed from one axis to another, as 

 in Fig. 244, where a a is the Urge bevil wheel, aud b b the 



147. 



The general law as to velocities of rotation and pres- 

 sure conveyed through bevil gearing, ia precisely the same 

 as in the case of plain gearing. 



RACKS. Gearing is sometimes used to convert a 

 rotary into a rectilineal motion, by the use of a rack and 

 pinion. 



The pinion a (Fig. 247) has teeth which fit between the 

 teeth of two rucks 6 and c ; and by giving o a reciprocat- 

 ing rotary motion round its centre, these two racks are 

 put in reciprocating rectilineal motion. 



The proper form for the teeth of a rack, so as to gear 

 with those of pinions formed as we have described, may 

 bo ascertained thus (Fig. 248) : In describing the teeth 

 of a wheel, the radius C F of the generating 

 circle bears a certain proportion to C B the 

 pitch radius ; or, as we have taken it, C F is 

 jjinds of C B. Again, FB, the radius of the 

 side of the tooth, is Jth of C B, or bears also 

 a constant ratio to C B. Hence, for every 

 wheel, B F makes a constant angle F B 

 with A C. Apply this to a rack N G, the 

 radius of the tooth must be at the same angle 

 to D E as that of F U to B C, or the anylo 

 N G E must be equal to the angle F B C ; but as the 

 jiitch line of the rack is a straight line, or may be sup- 

 posed a circle of infinitely great radius, so the radius N ( ' 

 of the tooth, being th of the pitch radius, must also be 

 infinitely great, ana the portion of the circular side of 

 the tooth described with this infinite radius must be a 

 straight line H G perpendicular to N G. To lind the 

 amount of obliquity or inclination of H G to D E, we 

 observe that it is the same with the inclination of B F to 

 B M (which is perpendicular to A C). Now M F is very 

 nearly bisected in L ; and as F L is Jth of F B, F M is 



218. 



-C 



smaller one, or the btril pinion ; or the 



>n may be communicated at any 



other angle, and in various directions 



Fig. 346. 



by cmiibmal.ons of bevil wheels and pinions, as in 



-45, whore the pinion a drives the wheel bb 



mated oWiqiMly to it ; and the pinion e, fixed on the 



f 6 b, drives the wheel d d at some other angle 



wild it. 



When the wheels aro equal, and flu ir axes at right 



H ""'iiT^f' "! '" Fig '-'"'' "' 

 Mhucally called mitre wheel*. 



very nearly jths^or jth of F B. So, in the rack, II K 

 must be Jth of K G ; and hence we have a simple mode 

 of getting out H G, the side of the rack tooth, by taking 

 G K any length, such as 1 inch, along the line G D, and 

 measuring up an offset K H = |th of K G. The line 

 H G is the side of the tooth, of which the top and bot- 

 tom may bo determined ;is in wheel gearing. 



WOliM AND WHEEL. There still remains a kind 

 of gearing most nj>plir:il>le in cases where it is desired to 

 reduce very greatly flu 1 angular velocity, or to increase 

 the moving pressure. We allude to the perpetual screw, 

 or worm and u-herl, as it is usually called (Fig. JM!i) : 

 '' l> the driving shaft has a screw or worm cut on its 

 oytindricd surface, the coils of which fit between Uvth 

 on the driven wheel a a. These teoth have sides in- 



